Hall mobilities and sheet carrier densities in a single ${\mathrm{Li}\mathrm{Nb}\mathrm{O}}_3$ conductive ferroelectric domain wall

In the last decade, conductive domain walls (CDWs) in single crystals of the uniaxial model ferroelectric lithium niobate (${\mathrm{Li}\mathrm{Nb}\mathrm{O}}_3$; LNO) have been shown to reach resistances more than 10 orders of magnitude lower than the resistance of the surrounding bulk, with charge carriers being firmly confined to sheets with a width of a few nanometers. LNO is thus currently witnessing increased attention because of its potential in the design of room-temperature nanoelectronic circuits and devices based on such CDWs. In this context, the reliable determination of the fundamental transport parameters of LNO CDWs, in particular the 2D charge carrier density $n_{2\mathrm{D}}$ and the Hall mobility ${\mu }_H$ of the majority carriers, is of great interest. In this contribution, we present and apply a robust and easy-to-prepare Hall-effect measurement setup by adapting the standard four-probe van der Pauw method to contact a single, hexagonally shaped domain wall that fully penetrates the 200-$\text{μ}\mathrm{m}$-thick LNO bulk single crystal. We then determine $n_{2\mathrm{D}}$ and ${\mu }_H$ for a set of external magnetic fields $B$ and prove the expected cosinelike angular dependence of the Hall voltage. Lastly, we present photoinduced-Hall-effect measurements of one and the same DW, by determining the impact of super-band-gap illumination on $n_{2\mathrm{D}}$.

Figure 1

(a),(c) Three-dimensional-CSHG-microscopy data and (b),(d) chromium-electrode arrangement 1–4 for the two samples ${\mathrm{LNO}}_1$ and ${\mathrm{LNO}}_2$ as prepared from $z$-cut ${\mathrm{Li}\mathrm{Nb}\mathrm{O}}_3$ single crystals for Hall-transport measurements using the van der Pauw method. Note that contacts 1–4 enclose a parallel junction of two CDWs, one at the front (dashed red line) and one at the back (dotted green line). The different protocols applied for DWC enhancement in ${\mathrm{LNO}}_1$ [(a),(b)] and ${\mathrm{LNO}}_2$ [(c),(d)] resulted in the different shapes and appearances as seen in (a),(c). The enhanced and desired head-to-head (H2H) DW inclination is color coded in red in the CSHG scale bar, while tail-to-tail- (T2T) type DWs appear in blue. As seen, sample ${\mathrm{LNO}}_2$ shows a larger DW inclination enhancement, justifying the larger DW current, as compared with sample ${\mathrm{LNO}}_1$.

Figure 2

Results of the macroscopic Hall-effect measurements for ${\mathrm{LNO}}_1$ and ${\mathrm{LNO}}_2$, performed as sketched in the inset. The magnetic field $B$ was applied perpendicular to the CDWs and then swept both positively and negatively. The charge carriers, i.e., mainly electrons—as discussed in the main text—that flow between current contacts 1 and 4 are deflected by the Lorentz force $F_L$, resulting in a measurable Hall voltage $U_H=U_{23}$ read between contacts 2 and 3. Plotted in the main diagram is the Hall resistance $R_h=U_{23}/I_{14}$ as a function of the magnetic field for the two samples, with ${\mathrm{LNO}}_2$ showing a nearly 10 times larger response. Note that the plotted $R_h$ values are the averaged values of $R_h(+B)$ and $R_h(-B)$ so as to compensate for sample misalignment effects (see the raw data in Fig. S5 and the reference $U_H$ measurement recorded from a monodomain bulk LNO crystal (containing no CDW at all) in Fig. S6(a) in Supplemental Material [28]). The corresponding 2D charge-carrier densities $n_{2\mathrm{D}}$ were then extracted from the slope of these linear relationships according to Eq. (1) and are discussed further in the main text and are summarized in Table 1.

Figure 3

(a) The DW of ${\mathrm{LNO}}_2$ was exemplarily rotated in the magnetic field of the electromagnet, with the angle $\mathrm{\Phi }$ being varied between $0^{\circ }$ and ${90}^{\circ }$ and the corresponding Hall voltages measured, as sketched in the inset. As expected, the Hall voltage $U_H$ decreases when $\mathrm{\Phi }$ is changed from $0^{\circ }$ to ${90}^{\circ }$ and the ratio $U_H/U_{H,0}$ with $U_{H,0}$ being the Hall voltage at $0^{\circ }$ (i.e., with the magnetic field vector being aligned perpendicular to both the CDW and the injected current) follows the theoretically predicted cosine function very clearly. All data points were acquired with a constant magnetic field of 400 mT. (b) Impact of super-band-gap illumination on the Hall transport for ${\mathrm{LNO}}_2$, which was placed in a constant magnetic field of $110\mathrm{mT}$ supplied by a permanent magnet and illuminated at a wavelength of 310 nm under a constant photon flux of ${10}^{13}{\mathrm{s}}^{-1}$. The Hall voltage $U_H=U_{23}$ was recorded as a function of the current $I=I_{14}$ for the two cases: (i) under illumination (purple data points) and (ii) in the dark (black data points). Both datasets show a linear behavior, however with a significantly increased response when the DW is illuminated, then generating additional electron-hole pairs.